Listing $a/(15-a)$ for $1\le a\le14,$ the integers are $2,4,14;$ the fractions with fractional part $1/2$ are $\tfrac12,\tfrac32,\tfrac{13}{2};$ and $\tfrac14,\tfrac{11}{4}$ have complementary fractional parts $1/4,3/4.$

The remaining fractional parts are $1/14,2/13,4/11,2/3,7/8,1/7,$ and none of their complements occurs. Thus only the integer, half-integer, and quarter cases can give integer sums.

Integer pairs give $4, 6, 8, 16, 18, 28.$ Half-integer pairs $\left(\tfrac12, \tfrac32, \tfrac{13}{2}\right)$ give $1, 2, 3, 7, 8, 13.$ The quarter pair $\tfrac14 + \tfrac{11}{4}$ gives $3.$

The distinct integers are $1, 2, 3, 4, 6, 7, 8, 13, 16, 18, 28,$ a total of $11.$

Thus, the correct answer is C.